Let $S=\{1, 2, 3, \ldots, 100\}$. Suppose $b$ and $c$ are chosen at random from the set $S$. The probability that $4x^2+bx+c=0$ has equal roots is

Which of the following six statements are true about the cubic polynomial $P(x) = 2x^3 + x^2 + 3x - 2$?
$(i)$ It has exactly one positive real root.
$(ii)$ It has either one or three negative roots.
$(iii)$ It has a root between $0$ and $1$.
$(iv)$ It must have exactly two real roots.
$(v)$ It has a negative root between $-2$ and $-1$.
$(vi)$ It has no complex roots.

If $\frac{1+\sqrt{3}i}{2}$ is a root of the equation $x^4-x^2+x-1=0$,then its real roots are:

Let $p, q \in \mathbb{Q}$. If $2 - \sqrt{3}$ is a root of the quadratic equation $x^2 + px + q = 0$,then:

In a triangle $ABC$,the value of $\angle A$ is given by $5\cos A + 3 = 0$. Find the quadratic equation whose roots are $\sin A$ and $\tan A$.

The polynomial equation of degree $5$ whose roots are the roots of the equation $x^5-3x^4-x^3+11x^2-12x+4=0$ each increased by $2$,is